Centre for Interdisciplinary Research in Computational Algebra (CIRCA) Researchhttps://hdl.handle.net/10023/1972024-03-29T10:51:49Z2024-03-29T10:51:49ZMaximal subsemigroups of finite transformation and diagram monoidsEast, JamesKumar, JitenderMitchell, James D.Wilson, Wilf A.https://hdl.handle.net/10023/171102024-02-17T00:42:18Z2018-06-15T00:00:00ZWe describe and count the maximal subsemigroups of many well-known transformation monoids, and diagram monoids, using a new unified framework that allows the treatment of several classes of monoids simultaneously. The problem of determining the maximal subsemigroups of a finite monoid of transformations has been extensively studied in the literature. To our knowledge, every existing result in the literature is a special case of the approach we present. In particular, our technique can be used to determine the maximal subsemigroups of the full spectrum of monoids of order- or orientation-preserving transformations and partial permutations considered by I. Dimitrova, V. H. Fernandes, and co-authors. We only present details for the transformation monoids whose maximal subsemigroups were not previously known; and for certain diagram monoids, such as the partition, Brauer, Jones, and Motzkin monoids. The technique we present is based on a specialised version of an algorithm for determining the maximal subsemigroups of any finite semigroup, developed by the third and fourth authors, and available in the Semigroups package for GAP, an open source computer algebra system. This allows us to concisely present the descriptions of the maximal subsemigroups, and to clearly see their common features.
The first author gratefully acknowledges the support of the Glasgow Learning, Teaching, and Research Fund in partially funding his visit to the third author in July, 2014. The second author wishes to acknowledge the support of research initiation grant [0076|2016] provided by BITS Pilani, Pilani. The fourth author wishes to acknowledge the support of his Carnegie Ph.D. Scholarship from the Carnegie Trust for the Universities of Scotland.
2018-06-15T00:00:00ZEast, JamesKumar, JitenderMitchell, James D.Wilson, Wilf A.We describe and count the maximal subsemigroups of many well-known transformation monoids, and diagram monoids, using a new unified framework that allows the treatment of several classes of monoids simultaneously. The problem of determining the maximal subsemigroups of a finite monoid of transformations has been extensively studied in the literature. To our knowledge, every existing result in the literature is a special case of the approach we present. In particular, our technique can be used to determine the maximal subsemigroups of the full spectrum of monoids of order- or orientation-preserving transformations and partial permutations considered by I. Dimitrova, V. H. Fernandes, and co-authors. We only present details for the transformation monoids whose maximal subsemigroups were not previously known; and for certain diagram monoids, such as the partition, Brauer, Jones, and Motzkin monoids. The technique we present is based on a specialised version of an algorithm for determining the maximal subsemigroups of any finite semigroup, developed by the third and fourth authors, and available in the Semigroups package for GAP, an open source computer algebra system. This allows us to concisely present the descriptions of the maximal subsemigroups, and to clearly see their common features.Computing maximal subsemigroups of a finite semigroupDonoven, C. R.Mitchell, J. D.Wilson, W. A.https://hdl.handle.net/10023/170722024-02-15T00:46:30Z2018-07-01T00:00:00ZA proper subsemigroup of a semigroup is maximal if it is not contained in any other proper subsemigroup. A maximal subsemigroup of a finite semigroup has one of a small number of forms, as described in a paper of Graham, Graham, and Rhodes. Determining which of these forms arise in a given finite semigroup is difficult, and no practical mechanism for doing so appears in the literature. We present an algorithm for computing the maximal subsemigroups of a finite semigroup S given knowledge of the Green's structure of S, and the ability to determine maximal subgroups of certain subgroups of S, namely its group H-classes. In the case of a finite semigroup S represented by a generating set X, in many examples, if it is practical to compute the Green's structure of S from X, then it is also practical to find the maximal subsemigroups of S using the algorithm we present. In such examples, the time taken to determine the Green's structure of S is comparable to that taken to find the maximal subsemigroups. The generating set X for S may consist, for example, of transformations, or partial permutations, of a finite set, or of matrices over a semiring. Algorithms for computing the Green's structure of S from X include the Froidure–Pin Algorithm, and an algorithm of the second author based on the Schreier–Sims algorithm for permutation groups. The worst case complexity of these algorithms is polynomial in |S|, which for, say, transformation semigroups is exponential in the number of points on which they act. Certain aspects of the problem of finding maximal subsemigroups reduce to other well-known computational problems, such as finding all maximal cliques in a graph and computing the maximal subgroups in a group. The algorithm presented comprises two parts. One part relates to computing the maximal subsemigroups of a special class of semigroups, known as Rees 0-matrix semigroups. The other part involves a careful analysis of certain graphs associated to the semigroup S, which, roughly speaking, capture the essential information about the action of S on its J-classes.
The third author wishes to acknowledge the support of his Carnegie Ph.D. Scholarship from the Carnegie Trust for the Universities of Scotland.
2018-07-01T00:00:00ZDonoven, C. R.Mitchell, J. D.Wilson, W. A.A proper subsemigroup of a semigroup is maximal if it is not contained in any other proper subsemigroup. A maximal subsemigroup of a finite semigroup has one of a small number of forms, as described in a paper of Graham, Graham, and Rhodes. Determining which of these forms arise in a given finite semigroup is difficult, and no practical mechanism for doing so appears in the literature. We present an algorithm for computing the maximal subsemigroups of a finite semigroup S given knowledge of the Green's structure of S, and the ability to determine maximal subgroups of certain subgroups of S, namely its group H-classes. In the case of a finite semigroup S represented by a generating set X, in many examples, if it is practical to compute the Green's structure of S from X, then it is also practical to find the maximal subsemigroups of S using the algorithm we present. In such examples, the time taken to determine the Green's structure of S is comparable to that taken to find the maximal subsemigroups. The generating set X for S may consist, for example, of transformations, or partial permutations, of a finite set, or of matrices over a semiring. Algorithms for computing the Green's structure of S from X include the Froidure–Pin Algorithm, and an algorithm of the second author based on the Schreier–Sims algorithm for permutation groups. The worst case complexity of these algorithms is polynomial in |S|, which for, say, transformation semigroups is exponential in the number of points on which they act. Certain aspects of the problem of finding maximal subsemigroups reduce to other well-known computational problems, such as finding all maximal cliques in a graph and computing the maximal subgroups in a group. The algorithm presented comprises two parts. One part relates to computing the maximal subsemigroups of a special class of semigroups, known as Rees 0-matrix semigroups. The other part involves a careful analysis of certain graphs associated to the semigroup S, which, roughly speaking, capture the essential information about the action of S on its J-classes.Proof-carrying plansSchwaab, Christopher JosephKomendantskaya, EkaterinaHill, AlisdairFarka, FrantišekPetrick, RonaldWells, JoeHammond, Kevinhttps://hdl.handle.net/10023/168552023-04-19T00:43:25Z2019-01-01T00:00:00ZIt is becoming increasingly important to verify safety and security of AI applications. While declarative languages (of the kind found in automated planners and model checkers) are traditionally used for verifying AI systems, a big challenge is to design methods that generate verified executable programs. A good example of such a “verification to implementation” cycle is given by automated planning languages like PDDL, where plans are found via a model search in a declarative language, but then interpreted or compiled into executable code in an imperative language. In this paper, we show that this method can itself be verified. We present a formal framework and a prototype Agda implementation that represent PDDL plans as executable functions that inhabit types that are given by formulae describing planning problems. By exploiting the well-known Curry-Howard correspondence, type-checking then automatically ensures that the generated program corresponds precisely to the specification of the planning problem.
2019-01-01T00:00:00ZSchwaab, Christopher JosephKomendantskaya, EkaterinaHill, AlisdairFarka, FrantišekPetrick, RonaldWells, JoeHammond, KevinIt is becoming increasingly important to verify safety and security of AI applications. While declarative languages (of the kind found in automated planners and model checkers) are traditionally used for verifying AI systems, a big challenge is to design methods that generate verified executable programs. A good example of such a “verification to implementation” cycle is given by automated planning languages like PDDL, where plans are found via a model search in a declarative language, but then interpreted or compiled into executable code in an imperative language. In this paper, we show that this method can itself be verified. We present a formal framework and a prototype Agda implementation that represent PDDL plans as executable functions that inhabit types that are given by formulae describing planning problems. By exploiting the well-known Curry-Howard correspondence, type-checking then automatically ensures that the generated program corresponds precisely to the specification of the planning problem.Automatic generation and selection of streamlined constraint models via Monte Carlo search on a model latticeSpracklen, PatrickAkgun, OzgurMiguel, Ian Jameshttps://hdl.handle.net/10023/158942024-03-27T00:38:28Z2018-01-01T00:00:00ZStreamlined constraint reasoning is the addition of uninferred constraints to a constraint model to reduce the search space, while retaining at least one solution. Previously it has been established that it is possible to generate streamliners automatically from abstract constraint specifications in Essence and that effective combinations of streamliners can allow instances of much larger scale to be solved. A shortcoming of the previous approach was the crude exploration of the power set of all combinations using depth and breadth first search. We present a new approach based on Monte Carlo search over the lattice of streamlined models, which efficiently identifies effective streamliner combinations.
Funding: EPSRC EP/P015638/1.
2018-01-01T00:00:00ZSpracklen, PatrickAkgun, OzgurMiguel, Ian JamesStreamlined constraint reasoning is the addition of uninferred constraints to a constraint model to reduce the search space, while retaining at least one solution. Previously it has been established that it is possible to generate streamliners automatically from abstract constraint specifications in Essence and that effective combinations of streamliners can allow instances of much larger scale to be solved. A shortcoming of the previous approach was the crude exploration of the power set of all combinations using depth and breadth first search. We present a new approach based on Monte Carlo search over the lattice of streamlined models, which efficiently identifies effective streamliner combinations.Using metric space indexing for complete and efficient record linkageAkgün, ÖzgürDearle, AlanKirby, Graham Njal CameronChristen, Peterhttps://hdl.handle.net/10023/151812024-02-15T00:36:57Z2018-01-01T00:00:00ZRecord linkage is the process of identifying records that refer to the same real-world entities in situations where entity identifiers are unavailable. Records are linked on the basis of similarity between common attributes, with every pair being classified as a link or non-link depending on their similarity. Linkage is usually performed in a three-step process: first, groups of similar candidate records are identified using indexing, then pairs within the same group are compared in more detail, and finally classified. Even state-of-the-art indexing techniques, such as locality sensitive hashing, have potential drawbacks. They may fail to group together some true matching records with high similarity, or they may group records with low similarity, leading to high computational overhead. We propose using metric space indexing (MSI) to perform complete linkage, resulting in a parameter-free process combining indexing, comparison and classification into a single step delivering complete and efficient record linkage. An evaluation on real-world data from several domains shows that linkage using MSI can yield better quality than current indexing techniques, with similar execution cost, without the need for domain knowledge or trial and error to configure the process.
2018-01-01T00:00:00ZAkgün, ÖzgürDearle, AlanKirby, Graham Njal CameronChristen, PeterRecord linkage is the process of identifying records that refer to the same real-world entities in situations where entity identifiers are unavailable. Records are linked on the basis of similarity between common attributes, with every pair being classified as a link or non-link depending on their similarity. Linkage is usually performed in a three-step process: first, groups of similar candidate records are identified using indexing, then pairs within the same group are compared in more detail, and finally classified. Even state-of-the-art indexing techniques, such as locality sensitive hashing, have potential drawbacks. They may fail to group together some true matching records with high similarity, or they may group records with low similarity, leading to high computational overhead. We propose using metric space indexing (MSI) to perform complete linkage, resulting in a parameter-free process combining indexing, comparison and classification into a single step delivering complete and efficient record linkage. An evaluation on real-world data from several domains shows that linkage using MSI can yield better quality than current indexing techniques, with similar execution cost, without the need for domain knowledge or trial and error to configure the process.Knowledge-based interoperability for mathematical software systemsKohlhase, MichaelDe Feo, LucaMüller, DennisPfeiffer, Markus JohannesRabe, FlorianThiéry, NicolasVasilyev, VictorWiesing, Tomhttps://hdl.handle.net/10023/124912023-04-19T00:42:12Z2017-01-01T00:00:00ZThere is a large ecosystem of mathematical software systems. Individually, these are optimized for particular domains and functionalities, and together they cover many needs of practical and theoretical mathematics. However, each system specializes on one area, and it remains very difficult to solve problems that need to involve multiple systems. Some integrations exist, but the are ad-hoc and have scalability and maintainability issues. In particular, there is not yet an interoperability layer that combines the various systems into a virtual research environment (VRE) for mathematics. The OpenDreamKit project aims at building a toolkit for such VREs. It suggests using a central system-agnostic formalization of mathematics (Math-in-the-Middle, MitM) as the needed interoperability layer. In this paper, we conduct the first major case study that instantiates the MitM paradigm for a concrete domain as well as a concrete set of systems. Specifically, we integrate GAP, Sage, and Singular to perform computation in group and ring theory. Our work involves massive practical efforts, including a novel formalization of computational group theory, improvements to the involved software systems, and a novel mediating system that sits at the center of a star-shaped integration layout between mathematical software systems.
Funding: OpenDreamKit Horizon 2020 European Research Infrastructures project (#676541) and DFG project RA-18723-1 OAF.
2017-01-01T00:00:00ZKohlhase, MichaelDe Feo, LucaMüller, DennisPfeiffer, Markus JohannesRabe, FlorianThiéry, NicolasVasilyev, VictorWiesing, TomThere is a large ecosystem of mathematical software systems. Individually, these are optimized for particular domains and functionalities, and together they cover many needs of practical and theoretical mathematics. However, each system specializes on one area, and it remains very difficult to solve problems that need to involve multiple systems. Some integrations exist, but the are ad-hoc and have scalability and maintainability issues. In particular, there is not yet an interoperability layer that combines the various systems into a virtual research environment (VRE) for mathematics. The OpenDreamKit project aims at building a toolkit for such VREs. It suggests using a central system-agnostic formalization of mathematics (Math-in-the-Middle, MitM) as the needed interoperability layer. In this paper, we conduct the first major case study that instantiates the MitM paradigm for a concrete domain as well as a concrete set of systems. Specifically, we integrate GAP, Sage, and Singular to perform computation in group and ring theory. Our work involves massive practical efforts, including a novel formalization of computational group theory, improvements to the involved software systems, and a novel mediating system that sits at the center of a star-shaped integration layout between mathematical software systems.Two variants of the froidure-pin algorithm for finite semigroupsJonusas, JuliusMitchell, J. D.Pfeiffer, M.https://hdl.handle.net/10023/118792023-04-18T23:36:18Z2018-02-08T00:00:00ZIn this paper, we present two algorithms based on the Froidure-Pin Algorithm for computing the structure of a finite semigroup from a generating set. As was the case with the original algorithm of Froidure and Pin, the algorithms presented here produce the left and right Cayley graphs, a confluent terminating rewriting system, and a reduced word of the rewriting system for every element of the semigroup. If U is any semigroup, and A is a subset of U, then we denote by <A> the least subsemigroup of U containing A. If B is any other subset of U, then, roughly speaking, the first algorithm we present describes how to use any information about <A>, that has been found using the Froidure-Pin Algorithm, to compute the semigroup <A∪B>. More precisely, we describe the data structure for a finite semigroup S given by Froidure and Pin, and how to obtain such a data structure for <A∪B> from that for <A>. The second algorithm is a lock-free concurrent version of the Froidure-Pin Algorithm.
2018-02-08T00:00:00ZJonusas, JuliusMitchell, J. D.Pfeiffer, M.In this paper, we present two algorithms based on the Froidure-Pin Algorithm for computing the structure of a finite semigroup from a generating set. As was the case with the original algorithm of Froidure and Pin, the algorithms presented here produce the left and right Cayley graphs, a confluent terminating rewriting system, and a reduced word of the rewriting system for every element of the semigroup. If U is any semigroup, and A is a subset of U, then we denote by <A> the least subsemigroup of U containing A. If B is any other subset of U, then, roughly speaking, the first algorithm we present describes how to use any information about <A>, that has been found using the Froidure-Pin Algorithm, to compute the semigroup <A∪B>. More precisely, we describe the data structure for a finite semigroup S given by Froidure and Pin, and how to obtain such a data structure for <A∪B> from that for <A>. The second algorithm is a lock-free concurrent version of the Froidure-Pin Algorithm.Erwin Schrödinger and quantum wave mechanicsO'Connor, John J.Robertson, Edmund F.https://hdl.handle.net/10023/115432024-02-26T00:43:09Z2017-08-22T00:00:00ZThe fathers of matrix quantum mechanics believed that the quantum particles are unanschaulich (unvisualizable) and that quantum particles pop into existence only when we measure them. Challenging the orthodoxy, in 1926 Erwin Schrödinger developed his wave equation that describes the quantum particles as a packet of quantum probability amplitudes evolving in space and time. Thus, Schrödinger visualized the unvisualizable and lifted the veil that has been obscuring the wonders of the quantum world.
2017-08-22T00:00:00ZO'Connor, John J.Robertson, Edmund F.The fathers of matrix quantum mechanics believed that the quantum particles are unanschaulich (unvisualizable) and that quantum particles pop into existence only when we measure them. Challenging the orthodoxy, in 1926 Erwin Schrödinger developed his wave equation that describes the quantum particles as a packet of quantum probability amplitudes evolving in space and time. Thus, Schrödinger visualized the unvisualizable and lifted the veil that has been obscuring the wonders of the quantum world.Timing properties and correctness for structured parallel programs on x86-64 multicoresHammond, KevinBrown, Christopher MarkSarkar, Susmithttps://hdl.handle.net/10023/99352023-01-03T11:30:13Z2016-01-01T00:00:00ZThis paper determines correctness and timing properties for structured parallel programs on x86-64 multicores. Multicore architectures are increasingly common, but real architectures have unpredictable timing properties, and even correctness is not obvious above the relaxed-memory concurrency models that are enforced by commonly-used hardware. This paper takes a rigorous approach to correctness and timing properties, examining common locking protocols from first principles, and extending this through queues to structured parallel constructs. We prove functional correctness and derive simple timing models, and both extend for the first time from low-level primitives to high-level parallel patterns. Our derived high-level timing models for structured parallel programs allow us to accurately predict upper bounds on program execution times on x86-64 multicores.
2016-01-01T00:00:00ZHammond, KevinBrown, Christopher MarkSarkar, SusmitThis paper determines correctness and timing properties for structured parallel programs on x86-64 multicores. Multicore architectures are increasingly common, but real architectures have unpredictable timing properties, and even correctness is not obvious above the relaxed-memory concurrency models that are enforced by commonly-used hardware. This paper takes a rigorous approach to correctness and timing properties, examining common locking protocols from first principles, and extending this through queues to structured parallel constructs. We prove functional correctness and derive simple timing models, and both extend for the first time from low-level primitives to high-level parallel patterns. Our derived high-level timing models for structured parallel programs allow us to accurately predict upper bounds on program execution times on x86-64 multicores.Adult dental anxiety : recent assessment approaches and psychological management in a dental practice settingHumphris, Gerald MichaelSpyt, JamesHerbison, AliceKelsey, Tomhttps://hdl.handle.net/10023/88212024-03-06T00:42:38Z2016-05-01T00:00:00ZDental Anxiety of patients is a common feature of the everyday experience of dental practice. This article advocates the use of regular assessment of this psychological construct to assist in patient management. Various tools, such as the Modified Dental Anxiety Scale (MDAS), are available to monitor dental anxiety that are quick to complete and easy to interpret. Patient burden is low. A new mobile phone assessment system (DENTANX) is being developed for distribution. This application and other psychological interventions are being investigated to assist patients to receive dental care routinely.
2016-05-01T00:00:00ZHumphris, Gerald MichaelSpyt, JamesHerbison, AliceKelsey, TomDental Anxiety of patients is a common feature of the everyday experience of dental practice. This article advocates the use of regular assessment of this psychological construct to assist in patient management. Various tools, such as the Modified Dental Anxiety Scale (MDAS), are available to monitor dental anxiety that are quick to complete and easy to interpret. Patient burden is low. A new mobile phone assessment system (DENTANX) is being developed for distribution. This application and other psychological interventions are being investigated to assist patients to receive dental care routinely.Mapping parallel programs to heterogeneous CPU/GPU architectures using a Monte Carlo Tree SearchGoli, MehdiMcCall, JohnBrown, Christopher MarkJanjic, VladimirHammond, Kevinhttps://hdl.handle.net/10023/61572022-04-13T14:30:10Z2013-06-20T00:00:00ZThe single core processor, which has dominated for over 30 years, is now obsolete with recent trends increasing towards parallel systems, demanding a huge shift in programming techniques and practices. Moreover, we are rapidly moving towards an age where almost all programming will be targeting parallel systems. Parallel hardware is rapidly evolving, with large heterogeneous systems, typically comprising a mixture of CPUs and GPUs, becoming the mainstream. Additionally, with this increasing heterogeneity comes increasing complexity: not only does the programmer have to worry about where and how to express the parallelism, they must also express an efficient mapping of resources to the available system. This generally requires in-depth expert knowledge that most application programmers do not have. In this paper we describe a new technique that derives, automatically, optimal mappings for an application onto a heterogeneous architecture, using a Monte Carlo Tree Search algorithm. Our technique exploits high-level design patterns, targeting a set of well-specified parallel skeletons. We demonstrate that our MCTS on a convolution example obtained speedups that are within 5% of the speedups achieved by a hand-tuned version of the same application.
2013-06-20T00:00:00ZGoli, MehdiMcCall, JohnBrown, Christopher MarkJanjic, VladimirHammond, KevinThe single core processor, which has dominated for over 30 years, is now obsolete with recent trends increasing towards parallel systems, demanding a huge shift in programming techniques and practices. Moreover, we are rapidly moving towards an age where almost all programming will be targeting parallel systems. Parallel hardware is rapidly evolving, with large heterogeneous systems, typically comprising a mixture of CPUs and GPUs, becoming the mainstream. Additionally, with this increasing heterogeneity comes increasing complexity: not only does the programmer have to worry about where and how to express the parallelism, they must also express an efficient mapping of resources to the available system. This generally requires in-depth expert knowledge that most application programmers do not have. In this paper we describe a new technique that derives, automatically, optimal mappings for an application onto a heterogeneous architecture, using a Monte Carlo Tree Search algorithm. Our technique exploits high-level design patterns, targeting a set of well-specified parallel skeletons. We demonstrate that our MCTS on a convolution example obtained speedups that are within 5% of the speedups achieved by a hand-tuned version of the same application.Repeating history : execution replay for Parallel Haskell programsFerrerio, HenriqueJanjic, VladimirCastro, LauraHammond, Kevinhttps://hdl.handle.net/10023/58952023-04-19T00:38:26Z2013-01-01T00:00:00ZParallel profiling tools, such as ThreadScope for Parallel Haskell, allow programmers to obtain information about the performance of their parallel programs. However, the information they provide is not always sufficiently detailed to precisely pinpoint the cause of some per- formance problems. Often, this is because the cost of obtaining that information would be prohibitive for a complete program execution. In this paper, we adapt the well-known technique of execution replay to make it possible to simulate a previous run of a program. We ensure that the non-deterministic parallel behaviour of the application is prop- erly emulated while the deterministic functional code is run unmodified. In this way, we can gather additional data about the behaviour of a par- allel program by replaying some parts of it with more detailed profiling information. We exploit this ability to identify performance bottlenecks in a quicksort implementation, and to derive a version that gives better speedups on multicore machines.
2013-01-01T00:00:00ZFerrerio, HenriqueJanjic, VladimirCastro, LauraHammond, KevinParallel profiling tools, such as ThreadScope for Parallel Haskell, allow programmers to obtain information about the performance of their parallel programs. However, the information they provide is not always sufficiently detailed to precisely pinpoint the cause of some per- formance problems. Often, this is because the cost of obtaining that information would be prohibitive for a complete program execution. In this paper, we adapt the well-known technique of execution replay to make it possible to simulate a previous run of a program. We ensure that the non-deterministic parallel behaviour of the application is prop- erly emulated while the deterministic functional code is run unmodified. In this way, we can gather additional data about the behaviour of a par- allel program by replaying some parts of it with more detailed profiling information. We exploit this ability to identify performance bottlenecks in a quicksort implementation, and to derive a version that gives better speedups on multicore machines.Maximal subsemigroups of the semigroup of all mappings on an infinite setEast, J.Mitchell, James DavidPéresse, Y.https://hdl.handle.net/10023/57932023-04-18T09:45:45Z2015-03-01T00:00:00ZIn this paper we classify the maximal subsemigroups of the full transformation semigroup ΩΩ, which consists of all mappings on the infinite set Ω, containing certain subgroups of the symmetric group Sym (Ω) on Ω. In 1965 Gavrilov showed that there are five maximal subsemigroups of ΩΩ containing Sym (Ω) when Ω is countable, and in 2005 Pinsker extended Gavrilov's result to sets of arbitrary cardinality. We classify the maximal subsemigroups of ΩΩ on a set Ω of arbitrary infinite cardinality containing one of the following subgroups of Sym (Ω): the pointwise stabiliser of a non-empty finite subset of Ω, the stabiliser of an ultrafilter on Ω, or the stabiliser of a partition of Ω into finitely many subsets of equal cardinality. If G is any of these subgroups, then we deduce a characterisation of the mappings f, g ∈ ΩΩ such that the semigroup generated by G ∪ {f, g} equals ΩΩ.
2015-03-01T00:00:00ZEast, J.Mitchell, James DavidPéresse, Y.In this paper we classify the maximal subsemigroups of the full transformation semigroup ΩΩ, which consists of all mappings on the infinite set Ω, containing certain subgroups of the symmetric group Sym (Ω) on Ω. In 1965 Gavrilov showed that there are five maximal subsemigroups of ΩΩ containing Sym (Ω) when Ω is countable, and in 2005 Pinsker extended Gavrilov's result to sets of arbitrary cardinality. We classify the maximal subsemigroups of ΩΩ on a set Ω of arbitrary infinite cardinality containing one of the following subgroups of Sym (Ω): the pointwise stabiliser of a non-empty finite subset of Ω, the stabiliser of an ultrafilter on Ω, or the stabiliser of a partition of Ω into finitely many subsets of equal cardinality. If G is any of these subgroups, then we deduce a characterisation of the mappings f, g ∈ ΩΩ such that the semigroup generated by G ∪ {f, g} equals ΩΩ.On disjoint unions of finitely many copies of the free monogenic semigroupAbughazalah, NabilahRuskuc, Nikhttps://hdl.handle.net/10023/33412023-04-18T09:47:16Z2013-08-01T00:00:00ZEvery semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.
2013-08-01T00:00:00ZAbughazalah, NabilahRuskuc, NikEvery semigroup which is a finite disjoint union of copies of the free monogenic semigroup (natural numbers under addition) is finitely presented and residually finite.Interfacing Coq + SSReflect with GAPKomendantsky, VladimirKonovalov, AlexanderLinton, Stephen Alexanderhttps://hdl.handle.net/10023/31752023-04-18T09:46:36Z2012-09-19T00:00:00ZWe report on an extendable implementation of the communication interface connecting Coq proof assistant to the computational algebra system GAP using the Symbolic Computation Software Composability Protocol (SCSCP). It allows Coq to issue OpenMath requests to a local or remote GAP instances and represent server responses as Coq terms.
Presentation slides and preprint both provided by author. Preprint published in Electronic Notes in Theoretical Computer Science: Proceedings of the 9th International Workshop On User Interfaces for Theorem Provers (UITP10).
2012-09-19T00:00:00ZKomendantsky, VladimirKonovalov, AlexanderLinton, Stephen AlexanderWe report on an extendable implementation of the communication interface connecting Coq proof assistant to the computational algebra system GAP using the Symbolic Computation Software Composability Protocol (SCSCP). It allows Coq to issue OpenMath requests to a local or remote GAP instances and represent server responses as Coq terms.Green index in semigroups : generators, presentations and automatic structuresCain, A.J.Gray, RRuskuc, Nikhttps://hdl.handle.net/10023/27602023-04-18T09:42:52Z2012-01-01T00:00:00ZThe Green index of a subsemigroup T of a semigroup S is given by counting strong orbits in the complement S n T under the natural actions of T on S via right and left multiplication. This partitions the complement S nT into T-relative H -classes, in the sense of Wallace, and with each such class there is a naturally associated group called the relative Schützenberger group. If the Rees index ΙS n TΙ is finite, T also has finite Green index in S. If S is a group and T a subgroup then T has finite Green index in S if and only if it has finite group index in S. Thus Green index provides a common generalisation of Rees index and group index. We prove a rewriting theorem which shows how generating sets for S may be used to obtain generating sets for T and the Schützenberger groups, and vice versa. We also give a method for constructing a presentation for S from given presentations of T and the Schützenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity (for subsemigroups), finite presentability (for extensions) and finite Malcev presentability (in the case of group-embeddable semigroups).
2012-01-01T00:00:00ZCain, A.J.Gray, RRuskuc, NikThe Green index of a subsemigroup T of a semigroup S is given by counting strong orbits in the complement S n T under the natural actions of T on S via right and left multiplication. This partitions the complement S nT into T-relative H -classes, in the sense of Wallace, and with each such class there is a naturally associated group called the relative Schützenberger group. If the Rees index ΙS n TΙ is finite, T also has finite Green index in S. If S is a group and T a subgroup then T has finite Green index in S if and only if it has finite group index in S. Thus Green index provides a common generalisation of Rees index and group index. We prove a rewriting theorem which shows how generating sets for S may be used to obtain generating sets for T and the Schützenberger groups, and vice versa. We also give a method for constructing a presentation for S from given presentations of T and the Schützenberger groups. These results are then used to show that several important properties are preserved when passing to finite Green index subsemigroups or extensions, including: finite generation, solubility of the word problem, growth type, automaticity (for subsemigroups), finite presentability (for extensions) and finite Malcev presentability (in the case of group-embeddable semigroups).Unary FA-presentable semigroupsCain, Alan JamesRuskuc, NikThomas, R.M.https://hdl.handle.net/10023/23752024-03-04T00:40:53Z2012-06-08T00:00:00ZAutomatic presentations, also called FA-presentations, were introduced to extend nite model theory to innite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classication of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a dicult problem in general. A restricted problem, also of signicant interest, is to ask this question for unary automatic presentations: auto-matic presentations over a one-letter alphabet. This paper studies unary FA-presentable semigroups. We prove the following: Every unary FA-presentable structure admits an injective unary automatic presentation where the language of representatives consists of every word over a one-letter alphabet. Unary FA-presentable semigroups are locally nite, but non-nitely generated unary FA-presentable semigroups may be innite. Every unary FA-presentable semigroup satises some Burnside identity.We describe the Green's relations in unary FA-presentable semigroups. We investigate the relationship between the class of unary FA-presentable semigroups and various semigroup constructions. A classication is given of the unary FA-presentable completely simple semigroups.
2012-06-08T00:00:00ZCain, Alan JamesRuskuc, NikThomas, R.M.Automatic presentations, also called FA-presentations, were introduced to extend nite model theory to innite structures whilst retaining the solubility of interesting decision problems. A particular focus of research has been the classication of those structures of some species that admit automatic presentations. Whilst some successes have been obtained, this appears to be a dicult problem in general. A restricted problem, also of signicant interest, is to ask this question for unary automatic presentations: auto-matic presentations over a one-letter alphabet. This paper studies unary FA-presentable semigroups. We prove the following: Every unary FA-presentable structure admits an injective unary automatic presentation where the language of representatives consists of every word over a one-letter alphabet. Unary FA-presentable semigroups are locally nite, but non-nitely generated unary FA-presentable semigroups may be innite. Every unary FA-presentable semigroup satises some Burnside identity.We describe the Green's relations in unary FA-presentable semigroups. We investigate the relationship between the class of unary FA-presentable semigroups and various semigroup constructions. A classication is given of the unary FA-presentable completely simple semigroups.Growth rates for subclasses of Av(321)Albert, M.H.Atkinson, M.D.Brignall, RRuskuc, NikSmith, RWest, Jhttps://hdl.handle.net/10023/21372023-04-18T09:42:57Z2010-10-22T00:00:00ZPattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates.
2010-10-22T00:00:00ZAlbert, M.H.Atkinson, M.D.Brignall, RRuskuc, NikSmith, RWest, JPattern classes which avoid 321 and other patterns are shown to have the same growth rates as similar (but strictly larger) classes obtained by adding articulation points to any or all of the other patterns. The method of proof is to show that the elements of the latter classes can be represented as bounded merges of elements of the original class, and that the bounded merge construction does not change growth rates.Generators and relations for subsemigroups via boundaries in Cayley graphsGray, RRuskuc, Nikhttps://hdl.handle.net/10023/21312023-04-18T09:42:52Z2011-11-01T00:00:00ZGiven a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.
2011-11-01T00:00:00ZGray, RRuskuc, NikGiven a finitely generated semigroup S and subsemigroup T of S we define the notion of the boundary of T in S which, intuitively, describes the position of T inside the left and right Cayley graphs of S. We prove that if S is finitely generated and T has a finite boundary in S then T is finitely generated. We also prove that if S is finitely presented and T has a finite boundary in S then T is finitely presented. Several corollaries and examples are given.Finite groups are big as semigroupsDolinka, IgorRuskuc, Nikhttps://hdl.handle.net/10023/20042023-04-18T09:43:52Z2011-09-01T00:00:00ZWe prove that a finite group G occurs as a maximal proper subsemigroup of an infinite semigroup (in the terminology of Freese, Ježek, and Nation, G is a big semigroup) if and only if |G| ≥ 3. In fact, any finite semigroup whose minimal ideal contains a subgroup with at least three elements is big.
2011-09-01T00:00:00ZDolinka, IgorRuskuc, NikWe prove that a finite group G occurs as a maximal proper subsemigroup of an infinite semigroup (in the terminology of Freese, Ježek, and Nation, G is a big semigroup) if and only if |G| ≥ 3. In fact, any finite semigroup whose minimal ideal contains a subgroup with at least three elements is big.Presentations of inverse semigroups, their kernels and extensionsCarvalho, C.A.Gray, RRuskuc, Nikhttps://hdl.handle.net/10023/19982023-04-18T09:42:54Z2011-06-01T00:00:00ZLet S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups.
"Part of this work was done while Gray was an EPSRC Postdoctoral Research Fellow at the University of St Andrews, Scotland"
2011-06-01T00:00:00ZCarvalho, C.A.Gray, RRuskuc, NikLet S be an inverse semigroup and let π:S→T be a surjective homomorphism with kernel K. We show how to obtain a presentation for K from a presentation for S, and vice versa. We then investigate the relationship between the properties of S, K and T, focusing mainly on finiteness conditions. In particular we consider finite presentability, solubility of the word problem, residual finiteness, and the homological finiteness property FPn. Our results extend several classical results from combinatorial group theory concerning group extensions to inverse semigroups. Examples are also provided that highlight the differences with the special case of groups.Simple extensions of combinatorial structuresBrignall, RRuskuc, NikVatter, Vhttps://hdl.handle.net/10023/19972024-03-24T00:40:25Z2011-07-01T00:00:00ZAn interval in a combinatorial structure R is a set I of points which are related to every point in R \ I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes — this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: An arbitrary structure S of size n belonging to a class C can be embedded into a simple structure from C by adding at most f (n) elements. We prove such results when C is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than 2. The function f (n) in these cases is 2, ⌈log2(n + 1)⌉, ⌈(n + 1)/2⌉, ⌈(n + 1)/2⌉, ⌈log4(n + 1)⌉, ⌈log3(n + 1)⌉ and 1, respectively. In each case these bounds are the best possible.
2011-07-01T00:00:00ZBrignall, RRuskuc, NikVatter, VAn interval in a combinatorial structure R is a set I of points which are related to every point in R \ I in the same way. A structure is simple if it has no proper intervals. Every combinatorial structure can be expressed as an inflation of a simple structure by structures of smaller sizes — this is called the substitution (or modular) decomposition. In this paper we prove several results of the following type: An arbitrary structure S of size n belonging to a class C can be embedded into a simple structure from C by adding at most f (n) elements. We prove such results when C is the class of all tournaments, graphs, permutations, posets, digraphs, oriented graphs and general relational structures containing a relation of arity greater than 2. The function f (n) in these cases is 2, ⌈log2(n + 1)⌉, ⌈(n + 1)/2⌉, ⌈(n + 1)/2⌉, ⌈log4(n + 1)⌉, ⌈log3(n + 1)⌉ and 1, respectively. In each case these bounds are the best possible.